A metal box with a square base and vertical sides is to contain 1024 cm3 of water, the material for the top and bottom costs Rs 5 per cm2 and the material for the sides costs Rs 2.50 per cm2. Find the least cost of the box.

Answer

Rs 1920

Explanation

Given the volume of the box = 1024 `cm^(3)`
Let length of the side of square base be x cm and height of the box be y ncm

`therefore` volume of the box `v=x^(2)y = 1024`
`rarr y=(1024)/(x^(2))`
Let c denote the cost of the box
`c=2x^(2)xx5+4xyxx2.50`
`=10x^(2)+10xy`
`=10x(x+y)`
`=10x(x+(1024)/(x^(2)))`
on differntiating both sides wrt x we get
`(dC)/(dx)=20x-(10240)/(x^(2))`
`(dC)/(dx)=0`
`rarr 20x=(10240)/(x^(2))`
`rarr20x^(3)=10240`
`rarr x^(3)=512=8^(3)`
x=8 ltrbgt Also `(d^(2)c)/(dx^(2))=20+(20480)/(x^(3))gt0`
`therefore` x=8 point of minima
Thus for x =8 cost is minimum and athe corresponding least cost of the box
`c(8)=10(8)^(2)+(10240)/(8)=640+1280=1920`
`therefore` Least cost =Rs 1920